Edge version of some degree based topological descriptors of graphs

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Journal of Mathematical Nanoscience

Available Online at: http://jmathnano.sru.ac.ir

Edge Version of Some Degree Based Topological Descriptors

of Graphs

K. Pattabiraman

∗

, T. Suganya

Department of Mathematics, Annamalai University, Annamalainagar 608 002, India.

Academic Editor: Saeid Alikhani

Abstract. In this paper, we study the edge version of some degree based topological indices such as general sum-connectivity index, Randic index, inverse sum indeg index, symmetric division deg index, augmenting Zagreb index and harmonic polynomial for joint graphs and certain graph operations.

Keywords. topological index, line graph, joint graph.

1 Introduction

A topological index is a mathematical measure which correlates to the chemical structures of any simple finite graph. They are invariant under the graph isomorphism. They play an important role in the study of QSAR/QSPR. There are numerous topological descriptors that have some applications in theoretical chemistry. Among these topological descriptors the degree-based topological indices are of great importance. The first degree-based topolog-ical indices that were defined by Gutman and Trinajsti´c in [7] 1972, are the first and second Zagreb indices. These indices were originally defined as follows:

M₁(G) = _∑ u∈V(G)

(

d_G(u))2= ∑ uv∈E(G)

(

d_G(u) +d_G(v)) and M₂(G) = _∑ uv∈E(G)

d_G(u)d_G(v), whered_G(u)is the degree of a vertexuinG.

In a graph G, if the corresponding edges share a vertex in G , the line graph L(G) of a graph Gis considered as a graph with vertices of the edges in G, and it possesses two adja-∗_{Corresponding author (Email address: [email protected]).}

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cent vertices. Similarly, the degree of an edgee∈ E(G) is represented by the number of its adjacent vertices inV(L(G)). In this paper, we study the edge version of some degree based topological indices such as general sum-connectivity index, Randic index, inverse sum indeg index, symmetric division deg index, augmenting Zagreb index and harmonic polynomial for joint graphs and certain graph operations.

2 Preliminaries

2.1 Edge version of Atom-bond connectivity index

The atom-bond connectivity index was proposed by Estrada et al. [1] which is defined as

ABC(G) =∑_uv_∈_E(G)

√

dG(u)+dG(v)−2

dG(u)×dG(v) . For more details about ABCindex, see [2–5]. Referring

to the end vertex degree d_G(e)and d_G(f)of edges eand f in a line graph ofG, Farahani [6] proposed the edge version of atom-bond connectivity index which is defined aseABC(G) =

∑uv∈E(G)

√

d_L(G)(e)+dL(G)(f)−2

d_L(G)(e)×dL(G)(f) , wheredL(G)(e)is the degree of the edgeein the line graphL(G).

2.2 Edge version of sum-connectivity index index

Thesum-connectivity indexwas proposed by Zhou and Trinajsti´c [8] in 2009, which is de-fined as the sum over all the edges of the graph of the terms(dG(u) +dG(v))

−1

2 . This concept

was extended to the general sum-connectivity index in 2010 [9], which is defined as follows:

χα(G) = ∑

uv∈E(G)

(d_G(u) +d_G(v))α. The edge version of sum-connectivity index expressed as

eχα(G) = ∑ e,f∈E(L(G))

(d_L(G)(e) +dL(G)(f))α, where α is a real number. The sum-connectivity

index correlate well with theπ-electron energy of benzenoid hydrocarbons [17].

2.3 Edge version of Randic index

TheRandic index which is defined as the multiple over all the edges of the graph of the terms(d_G(u)d_G(v))−12 . The edge version of Randic index expressed as

eRα(G) =

∑

e,f∈E(L(G))

(d_L(G)(e)dL(G)(f))α.

2.4 Edge version of Harmonic index

Zhang [15, 16] introduced the harmonic index in 2012 which is defined as

H(G) =

∑

uv∈E(G)

2

dG(x) +dG(y)

.

The edge version of harmonic index is defined as

eH(G) =

∑

e f∈E(L(G))

2

d_L(G)(e) +dL(G)(f)

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2.5 Edge version of inverse sum indeg and symmetric division deg indices

The edge version of inverse sum indeg index is defined as

eISI(G) =

∑

e f∈E(L(G)

d_L(G)(e)dL(G)(f) d_L(G)(e) +dL(G)(f)

.

whered_L(G)(e)is the degree of the edgeeinL(G)and the edge version of symmetric division

deg index is defined aseSDD(G) = ∑ e f∈E(L(G)

d_L(G)(e)2+dL(G)(f)2

d_L(G)(e)dL(G)(f) .

2.6 Edge version of Augumenting Zagreb index index

Furtula et al [12] modify the ABC index and named as Augumenting Zagreb index. The correlating ability among several topological indices possess by AZI. The edge version of

AZI is defined aseAZI(G) = ∑ e f∈E(L(G)

( _d

L(G)(e)dL(G)(f)

d_L(G)(e)+dL(G)(f)−2

)3 .

2.7 Edge version of Harmonic polynomial

The harmonic polynomial is defined in [11] asH(G,x) = ∑ uv∈E(G)

2xdG(u)+dG(v)−1_{. Note that}

´₁

0 H(G,x)dx= H(G). The edge version of harmonic polynomial is defined as eH(G,x) = ∑

e f∈E(L(G)

2xdL(G)(e)+dL(G)(f)−1_.

3 Main Results

Joint structure and graph operation of basic molecular structures are frequently found in the new chemical compounds, nanomaterials and drugs in the fields of chemical and phar-maceutical engineering. The phenomenon provides us some hints on the significance and feasibility of the research on the chemical and pharmacological properties of these molecular structures. In this section, the edge version of some topological indices for joint graphs and certain graph operations are determined.

b

b b

b

b b b b

Figure 1. The graphPn+Cm.

Theorem 3.1. Let G=Pn+Cm be the join graph for n≥4,m≥3,see Figure 1. Then (i)eχα(Pn+Cm) = (n+m−7)(4)α+3α+3(5)α+3(6)α.

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Proof. (i) Apply induction method, one can see that this line graph possesses n+m

edges. Letd_L(G)(e)and dL(G)(f)be the degree of edge ofe. From the structure of the graph Pn+Cm, we consider the following.

Table 1.1. Degree of vertices in L(G).

(d_L(G)(e),dL(G)(f)), wheree f ∈E(L(G)) Number of edges

(1,2) 1

(2,2) n+m−7

(2,3) 3

(3,3) 3

From the definition ofeχαand Table 1.1, we have

eχα(G) =

∑

e,f∈E(L(G))

(d_L(G)(e) +dL(G)(f))α

= (n+m−7)(4)α+3α+3(5)α+3(6)α.

A similar argument of(i), we obtain the result of(ii).

Corollary 3.2. Let G=Pn+Cm be the join graph. Then

eχα(G) =

                

4(n+m+2), if α=1; 1

60(15n+15m+71), if α=−1; 16(n+m+5), if α=2;

2(n+m−7) +√3+3√5+3√6, if α= 1₂;

n+m−7

2 +√3₅+√3₆+√1₃+, if α=−21.

Proof: Puttingα=1 in Theorem 2.1, we get

eχ(G) =3+ (n+m−1)4+15+18 =4(n+m+2).

Puttingα=−1 in Theorem 2.1, we get

eχ−1(G) = (n+m−7)(4)−1+3−1+3(5)−1+3(6)−1

= 1

60(15n+15m+71). Puttingα=2 in Theorem 2.1, we get

eχ2(G) = (n+m−7)(4)2+32+3(5)2+3(6)2

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Puttingα=1₂ in Theorem 2.1, we get

eχ1

2(G) = (n+m−7)(4) 1

2 ₊₃12 ₊₃₍₅₎12 ₊₃₍₆₎12

=2(n+m−7) +√3+3√5+3√6. Puttingα= −₂1 in Theorem 2.1, we get

eχ−1

2 (G) = (n+m−7)(4) −1

2 +3−12 +3(5)−12 +3(6)−12

= √1

3+

n+m−7

2 +

3

√

5 + 3

√

6.

Corollary 3.3. Let G=Pn+Cm be the join graph. Then

eRα(G) =

                

4n+4m+19, if α=1; 1

12(3n+3m−5), if α=−1; 16n+16m+243, if α=2;

2(n+m) +√2+3√6−5, if α= 1₂;

n+m−5

2 + √1₂ +√3₆, if α= −21.

Using Table 1.1 and definitions of the edge version of ISI SDD AZI and eH(G,x), we

obtain the following theorem.

Theorem 3.4. Let G=Pn+Cm be the join graph. Then (i)eSDD(G) =2(n+m)−1.

(ii)eISI(G) =n+m+53₃₀.

(iii) eAZI(G) = ₁₆1(128n+128m−141).

(iv)eH(G,x) =2x2(1+x(n+m−7)) +6x4(1+x).

b

b b

b

b b

b

b b b

b

Figure 2. The graphPn+Sm.

Theorem 3.5. Let G=Pn+Sm be the join graph for n,m≥4,see Figure 2. Then

(i)eχα(G) =3α+ (n−4)4α+ (2+m)α+(m−1)(₂m−2)(2(m−1))α+ (m−1)(2m−1)α. (ii)eRα(G) =2α+4α(n−4) + (2m)α+(m−1)(₂m−2)((m−1)2)α+ (m−1)(m2−m)α. Proof. Apply the induction method, one can see that this line graph possesses

2(m+2) + (m−1)(m−2)

2

edges. Let d_L(G)(e)be the degree of edge of e. From the structure of the graph Pn +Sm, we

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(1,2) 1

(2,2) n−4

(2,m) 1

(m−1,m−1) (m−1)(₂m−2)

(m−1,m) m−1

From the definition ofeχαand Table 1.2, we obtain

eχα(G) =

∑

e,f∈E(L(G))

=3α+ (n−4)4α+ (2+m)α+ (m−1)(m−2)

2 (2(m−1))

α

+ (m−1)(2m−1)α. A similar argument of(i), we get(ii).

Corollary 3.6. Let G=Pn+Sm be the join graph. Table 1.2, we obtain

eχα(G) =

                  

m3−2m2+3m+4n−8, if α=1;

n+₂₊1_m +(m−1)(m−2)

4(m−1) + m−1

2m−1, if α=−1; 2m4−6m3+25m2+13m+16n−40, if α=2;

√

m+2+m(m₂−1)√(2m−1) +√3+2(n−4), if α= 1₂; 1

√

2+m +

(m−1)(m−2)

2√(2(m−1)) +

(m−1)

√

(2m−1) +

1

√

3+

(n−4)

2 , if α= −21.

Proof.Puttingα=1 in Theorem 2.7, we get

eχ(G) =3+ (n−4)4+ (2+m) +(m−1)(m−2 )

2 (2(m−1)) + (m−1)(2m−1)

=m3−2m2+3m+4n−8. Puttingα=−1 in Theorem 2.7, we get

eχ−1(G) =3−1+ (n−4)4−1+ (2+m)−1+

(m−1)(m−2)

2 (2(m−1))

−1

+ (m−1)(2m−1)−1=n+ 1

2+m +

(m−1)(m−2)

4(m−1) +

m−1 2m−1. Puttingα=2 in Theorem 2.7, we get

eχ2(G) =32+ (n−4)42+ (2+m)2+ (m−1)(m−2

)

2 (2(m−1))

2_{+ (m}₋₁₎₍₂_m₋₁₎2

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eχ1

2(G) =3

2_{+ (n}₋₄₎₄1₂ _{+ (}₂₊_m)1₂ ₊ (m−1)(m−2)

2 (2(m−1))

1

2 + (m−1)(2m−1)12

=√3+2(n−4) +√2+m+(m−1)(m−2)

2

√

(2(m−1)) + (m−1)

√

(2m−1). Puttingα= −₂1 in Theorem 2.7, we get

eχ−1

2 (G) =3

2_{+ (n}₋₄₎₄−1₂ _{+ (}₂₊_m)−1₂ ₊(m−1)(m−2)

2 (2(m−1))

−1

2 + (m−1)(2m−1)−12

= √1

3+

(n−4)

2 +

1

√

2+m +

(m−1)(m−2)

2√(2(m−1)) +

(m−1)

√

(2m−1). Corollary 3.7. Let G=Pn+Sm be the join graph. Then

eRα(G) =

                       1

2(m4−3m3−5m2−m+8n+26), if α=1;

(m−2)

2(m−1) + (m−1)

(m2₋_m₎ +₂1_m + n−₄4 +1₂, if α=−1;

(m−1)(m−2)

2 (m2−2m+1)2+ (m−1)(m4+m2−2m3) +4m2+4

+16(n−4), if α=2;

m3₋₄_m₊₅_m

2 +

√

2m+2n+√2−9+ (m−1)√m2−_m_, _if _α₌ 1 2;

m−1

√

m2₋_m +

1

√

2m +

n+m−6

2 +√1₂, if α= −21.

Proof: Puttingα=1 in Theorem 2.9, we get

eR(G) =2+4(n−4) + (2m) + (m−1)(m−2 )

2 ((m−1)

2_{) + (m}₋₁_)(m2₋_m)

=1

2(m

4₋₃_m3₋₅_m2₋_m₊₈_n₊₂₆₎_.

Puttingα=−1 in Theorem 2.9, we get

eR−1(G) =2−1+4−1(n−4) + (2m)−1+

(m−1)(m−2)

2 ((m−1)

2₎−1_{+ (m}₋₁_)(m2₋_m)−1

= 1

2+

n−4

4 +

1 2m+

(m−1)(m−2)

2(m−1)2 +

(m−1) m2₋_m. Puttingα=2 in Theorem 2.9, we get

eR2(G) =22+42(n−4) + (2m)2+

(m−1)(m−2)

2 ((m−1)

2₎2_{+ (m}₋₁_)(m2₋_m)2

= (m−1)(m−2)

2 (m

2₋₂_m₊₁₎2_{+ (m}₋₁_)(m4₌_m2₋₂_m3_{) +}₄_m2₊₄₊₁₆_(n₋₄₎_.

eR1

2(G) =2 1

2 ₊₄12_(n−₄_{) + (}₂_m)12 ₊(m−1)(m−2)

2 ((m−1)

2₎1₂ _{+ (m}₋₁_)(m2₋_m)1₂

= m

3₋₄_m₊₅_m

2 +

√

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Puttingα= −₂1 in Theorem 2.9, we get

eR−1

2 (G) =2 −1

2 ₊₄−12 _(n−₄_{) + (}₂_m)−12 ₊(m−1)(m−2)

2 ((m−1)

2₎−1₂ _{+ (m}₋₁_)(m2₋_m)−1₂

= √1

2+

n−4

2 +

1

√

2m +

(m−1)(m−2)

2(m−1) +

m−1

√

m2−_m.

Using Table 1.2 and definitions of the edge version of ISI, SDD, AZI and H(G,x), we obtain the following theorem.

Theorem 3.8. Let G=Pn+Sm be the join graph. Then (i)eSDD(G) = 6n−₃19 +m

2₊₄

4m +m

4₋₅_m3₊₉_m2₋₇_m₊₂

2(m−1) +2m

3₋₄_m2₊₃_m₋₁

2m−1 ,

(ii)eISI(G) = (m−1)

2₍_m₋₂₎

4 +m

3₋₂_m2₊_m

2m−1 +22+mm +3n−310,

(iii) eAZI(G) =8n−16+(m−1)

7₍_m₋₂₎

16m3 +

(m−1)4m3 (m2₋_m₋₂₎3.

(iv)eH(G,x) = (n−4)2x3+2x2+2x(m−1)+ (m−1)(m−2)x2m−1+ (m−1)2x2m.

b

b b

b

b b

b

b b

b

b b b b b b b

Figure 3. The graphCm +Pn+Cm.

Theorem 3.9. Let G=Cm+Pn +Cm be the join graph for n≥4,m≥3,see Figure 3. Then

(i)The edge version of general sum connectivity index is eχα(G) = (n+2m−10)(4)α+6(5)α+

6(6)α.

(ii)The edge version of Randic index iseRα(Cm+Pn+Cm) =2α+4α+6(6)α+6(9)α.

Proof. Apply the induction method, one can see that this line graph possesses n+2m+2 edges. Letd_L(G)(e)and dL(G)(f)be the degree of edge ofe. From the structure of the graph Cm+Pn+Cm, we consider the following degree cases

(2,2) n+m−10

(2,3) 6

(3,3) 6

From the definition ofeχαand Table 1.3, we have eχα(G) =

∑

e,f∈E(L(G))

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Corollary 3.10. Let G=Cm+Pn +Cm be the join graph. Then

eχα(G) =

                

4n−8m+26, if α=1; 5n+10m−6

20 , if α=−1; 16n+32m+206, if α=2;

(2n+2m−10) +6√5+6√6, if α=1₂;

n+2m−10

2 +√6₅+√6₆, if α= −21.

Corollary 3.11. Let G=Cm+Pn +Cm be the join graph. Then

eRα(G) =

                

2(2n+4m+5), if α=1; 1

12(3n+6m−10), if α=−1; 16n+32m+542, if α=2; 2n+4m−2+2√6, if α=1₂;

n+2m−6+2√6

2 , if α= −21.

Using Table 1.3 and definitions of the edge version of the indices ISI, SDD, AZI and

H(G,x), we obtain the following theorem.

Theorem 3.12. Let G=Cm+Pn +Cm be the join graph. Then (i)eSDD(G) =2n+4m+5.

(ii)eISI(G) =n+2m+31₅. (iii) eAZI(G) =2n+22m.

(iv)eH(G,x) = (n+2m)2x3−4x3(2x2+2x+5).

b

b b b b

b

b b

b

bb

b b b

Figure 4. The graphSm+Pn+Sm.

b

Theorem 3.13. Let G=Sm+Pn+Smbe the join graph for n,m≥4,see Figure 4. Then (i)eχα(G) = (n−4)4α+2(2+m)α+ (m−1)(m−2)(2(m−1))α+2(m−1)(2m−1)α.

(ii)eRα(G) =4α(n−4) +2(2m)α+ (m−1)(m−2)((m−1)2)α+2(m−1)(m2−m)α.

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Table 1.4. Degree of vertices inL(G).

(2,2) n−4

(2,m) 2

(m−1,m−1) (m−1)(m−2)

(m,m−1) 2(m−1)

From the definition ofeχα, and Table 1.4, we have

eχα(G) =

∑

e,f∈E(L(G))

= (n−4)4α+2(2+m)α+ (m−1)(m−2)(2(m−1))α +2(m−1)(2m−1)α.

A similar argument of(i)we get the result of(ii).

Corollary 3.14. Let G=Sm+Pn+Smbe the join graph. Then

eχα(G) =

                      

2(2n+m3−2m2+3m−7), if α=1;

(n−4)+2(m−2)

4 +2+2m +

2(m−1)

2m−1 , if α=−1; 2m4−2m3+30m2+4m−74, if α=2;

(n−4)2+2√2+m+ (m−1)(m−2)√2(m−1) +2(m−1)√2m−1, if α=1₂;

n−4

2 +√₂2+m +

(m−1)(m−2)

2√(m−1) +

2(m−1)

√

2m−1, if α=

−1 2 .

Corollary 3.15. Let G=Sm+Pn+Smbe the join graph. Then

eRα(G) =

                  

m4−3m3+5m2−m−14, if α=1;

n−4 4 + m

2₊_m₋₃

m(m−1), if α=−1;

16n+2m5+3m4−6m3+12m2−4m−63, if α=2;

(n−4)2+2√2m+ (m−1)2(m−2) +√2(m−1)m2−m, if α= 1₂; 2(m−1)

m2₋_m + (m−2) +√2₂_m +

(n−4)

4 , if α= −21.

Using Table 1.4 and definitions of the edge version of ISI, SDD, AZI and H(G,x), we obtain the following theorem.

Theorem 3.16. Let G=Sm+Pn+Smbe the join graph. Then (i)eSDD(G) =2(n−4) + 3m

4₋₉_m3₊₁₆_m2₋₁₀_m₊₁

m(n−1) . (ii)eISI(G) =n−4+_m4+m2+m

3₋₄_m2₊₅_m₋₂

2 +m

3₋_m2

2m−1 .

(iii) eAZI(G) =8n+ (m−1)4

(

(m−2)(m−1)2

2m−4 + 2m

3

2m−3 )

−16.

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b

b b b

b

b b

b b b

b

b b b

Figure 5. The graphCm+Pn+Sr.

b

Theorem 3.17. Let G=Cm+Pn +Sr be the join graph for n,r≥4,m≥3,see Figure5.Then (i)eχα(G) = (n+m−7)4α+3(5)α+r+1α+3(6)α+(r−1)(₂r−2)(2(r−1))α+ (r−1)(2r−1)α. (ii) eRα(Sm +Pn+Sr) = (n+m−7)4α+3(6)α+ (2(r−1))α+3(9)α+ (r−1)(₂r−2)(r−1)α+ (r−1)(r(r−1))α.

Proof. Apply induction method, one can see that this line graph possesses 2(n+m)+(₂ r−1)r edges. Letd_L(G)(e) anddL(G)(f)are the degree of edge ofe. From the structure of the graph Cm+Pn+Sr, we consider the following degree cases.

(2,2) n+m-7

(r−1,r−1) (r−1)(₂r−2)

(2,3) 3

(2,r−1) 1

(3,3) 3

(r−1,r) r−1

From the definitions ofeχα,eRα, and Table 1.4, we obtain the required results.

Corollary 3.18. Let G=Cm+Pn +Sr be the join graph. Then

eχα(G) =

                  

4(n+m) +r3−2r2+2r+4, if α=1;

n+m−7 4 +r

2₊₃_r₊₂

4(r+1) + (r−1)

2r−1 + 1110, if α=−1;

(n+m−7)16+2r4−6r3+11r2−7r+187, if α=2;

(n+m−7)2+3√5+√r+1+3√6+(r−₂1)n√2(r−1), if α=1₂;

n+m−7

2 + √3₅ +√r1+1 + 3

√

6+

(r−1)(r−2)

2√2(n−1) + (r−1)

√

2r−1, if α=

(12)

Corollary 3.19. Let G=Cm+Pn +Sr be the join graph. Then

eRα(G) =

                        

4(n+m) +3r3−7r2₂+9r+29, if α=1;

(r−1)(r−2)+1

2(r−1) +r(rr−−11) +n +m−7

4 +56, if α=−1;

(n+m)16+4(r2−2r) +r2(r−1)3+(r−1)₂3(r−2) +243, if α=2;

(n+m−7)2+(r−1)₂2(r−2) + (r−1)√r(r−1) +√2(r−1) +3√6+9, if α= 1₂;

(n+m−7)

2 +

(r−1)(r−2)

2√r−1 +

(r−1)

√

r(r−1) +

1

√

2(r−1) +

√

6+3

√

6 , if α=

−1 2 .

Corollary 3.20. Let G=Cm+Pn +Sr be the join graph. Then (i)eSDD(G) = (n+m)2+53₂ +4+(r−1)

2

2(r−1) + (r−1)(r−2) +

(r−1)3₊_r2

r(r−1) . (ii)eISI(G) =n+m+11₁₀ +₂2₊₍(r_r−₋1₁)₎ +(r−1)

2₍_r₋₂₎

4 +2r−r 1. (iii) eAZI(G) = (n+m)8+139₆₄ +8(r−2)

3

(r−1)3 +

(r−1)6

16(r−2) +

r3(r−1)4 (2r−3)3.

(iv)eH(G,x) = (n+m−7)2x3+6x4(1+x) +2xr+ (r−1)(r−2)x(2r−3)+2(r−1)x2(r−1).

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